##
**Methods of dynamic and nonsmooth optimization.**
*(English)*
Zbl 0696.49003

CBMS-NSF Regional Conference Series in Applied Mathematics, 57. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. v, 90 p. $ 15.75 (1989).

This is a nice booklet (90 pages) about the study and applications of the differential properties of functions and sets not differentiable in the usual sense. The term “nonsmooth analysis” was coined for this scientific area. The booklet contains four chapters of which the titles are, respectively, nonsmooth analysis and geometry; the basic problem in the calculus of variations; verification functions and dynamic programming; optimal control. For a first introduction into these fields the booklet may be less suitable, but the presentation, in a unified way, of the material will appeal to readers who already had some exposure to these fields before. The monograph (that is what the booklet is) shows subtle concepts and mathematical reasonings in an elegant way. The contents stresses more the unified point of view which combines the subjects dealt with in the chapters rather than that new tools for solving new or existing problems are presented. In fact, it is the reviewer’s impression that most of the examples and solutions shown have been around for some time.

According to the preface the monograph is based on a series of lectures given at a Conference in Atlanta, Georgia, in 1986; more recent results have been incorporated. Now a description of the chapters follows.

Chapter 1 starts with a simple motivating example of a nondifferentiable value function and its role as a Lagrange multiplier. Subsequently notions as perpendicular, proximal normal, cone, subdifferential, generalized Jacobian, are introduced. Subdifferential and directional derivatives are related to Lipschitz functions. Some calculus is given (what about the subdifferential of a sum of two functions?). The last sessions apply the concepts introduced to value functions.

Chapter 2, on the calculus of variations, starts with a brief discussion on the Euler equation, the necessary conditions of Weierstrass and Jacobi and the Erdmann condition. Within and after the section on Tonelli’s existence theorem there is a discussion on how crucial it is for the solution of the variational problem to consider the right class of candidate solutions (the class of continuously differentiable functions or of absolutely continuous functions or of Lipschitzfunctions or of...). A generalization of the Euler equation is given; it becomes an inclusion equation when the subdifferential plays an essential role. Further, Hamiltonian trajectories and regularity conditions are studied.

Whereas chapter 2 dealt with deductive methods in optimization (necessary conditions yield some candidate solutions; if there is more than one candidate further elimination is required), chapter 3 deals with inductive methods (identify a candidate solution through some means and prove that it is at least as good as any other solution). These induction methods lead to a complex of ideas around dynamic programming and the Hamiltonian-Jacobi equation. Thus it is not surprising that value functions play a dominant role.

Chapter 4, on optimal control, starts with a discussion of the maximum principle of Pontryagin, its relation to Hamiltonian systems and the Fenchel transform of convex analysis. Hamiltonian inclusion relations are defined for optimal control problems defined by means of differential inclusions. Other subjects in this chapter are among others, attainable sets and controllability, Hamiltonian multipliers, controllability of Hamiltonian flows on convex energy surfaces, problems with variable final time.

In conclusion, the monograph discusses “nonsmooth analysis” as a unified approach to the study of many different issues (including nondifferentiability issues specifically) in optimization and analysis. The style in which it is written is interest-arousing. The author touches upon many scientific disciplines in the only 90 pages of the book. Therefore the contents will probably be appreciated especially by those readers who are no novices in these disciplines.

According to the preface the monograph is based on a series of lectures given at a Conference in Atlanta, Georgia, in 1986; more recent results have been incorporated. Now a description of the chapters follows.

Chapter 1 starts with a simple motivating example of a nondifferentiable value function and its role as a Lagrange multiplier. Subsequently notions as perpendicular, proximal normal, cone, subdifferential, generalized Jacobian, are introduced. Subdifferential and directional derivatives are related to Lipschitz functions. Some calculus is given (what about the subdifferential of a sum of two functions?). The last sessions apply the concepts introduced to value functions.

Chapter 2, on the calculus of variations, starts with a brief discussion on the Euler equation, the necessary conditions of Weierstrass and Jacobi and the Erdmann condition. Within and after the section on Tonelli’s existence theorem there is a discussion on how crucial it is for the solution of the variational problem to consider the right class of candidate solutions (the class of continuously differentiable functions or of absolutely continuous functions or of Lipschitzfunctions or of...). A generalization of the Euler equation is given; it becomes an inclusion equation when the subdifferential plays an essential role. Further, Hamiltonian trajectories and regularity conditions are studied.

Whereas chapter 2 dealt with deductive methods in optimization (necessary conditions yield some candidate solutions; if there is more than one candidate further elimination is required), chapter 3 deals with inductive methods (identify a candidate solution through some means and prove that it is at least as good as any other solution). These induction methods lead to a complex of ideas around dynamic programming and the Hamiltonian-Jacobi equation. Thus it is not surprising that value functions play a dominant role.

Chapter 4, on optimal control, starts with a discussion of the maximum principle of Pontryagin, its relation to Hamiltonian systems and the Fenchel transform of convex analysis. Hamiltonian inclusion relations are defined for optimal control problems defined by means of differential inclusions. Other subjects in this chapter are among others, attainable sets and controllability, Hamiltonian multipliers, controllability of Hamiltonian flows on convex energy surfaces, problems with variable final time.

In conclusion, the monograph discusses “nonsmooth analysis” as a unified approach to the study of many different issues (including nondifferentiability issues specifically) in optimization and analysis. The style in which it is written is interest-arousing. The author touches upon many scientific disciplines in the only 90 pages of the book. Therefore the contents will probably be appreciated especially by those readers who are no novices in these disciplines.

Reviewer: G.Olsder